To solve the problem, we can use the definitions of the sine and cosine functions in relation to the sides of the right triangle.
In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse, and the cosine of an angle is defined as the ratio of the length of the adjacent side to the hypotenuse.
Given the triangle with legs measuring 3 and 4, we can find the hypotenuse (let's denote it as \(h\)) using the Pythagorean theorem:
\[ h = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Now, we can find \( \sin x° \) and \( \cos y° \):
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Finding \( \sin x° \):
- The angle \( x \) is opposite the leg measuring 3.
- The hypotenuse is 5.
- Therefore,
\[ \sin x° = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5} \]
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Finding \( \cos y° \):
- The angle \( y \) is opposite the leg measuring 4.
- The hypotenuse is still 5.
- Therefore,
\[ \cos y° = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} \]
Next, we examine the relationship between \( \sin x° \) and \( \cos y° \):
From the fundamental relationship of angles in a right triangle, we know that:
\[ x + y = 90° \quad \text{(because they sum to the right angle)} \]
Using the co-function identities, we have:
\[ \sin x° = \cos(90° - x) = \cos y° \]
Thus, \( \sin x° = \cos y° \).
In conclusion:
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The values are:
- \( \sin x° = \frac{3}{5} \)
- \( \cos y° = \frac{4}{5} \)
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Relationship: Since \( \sin x° \) and \( \cos y° \) represent the same angle due to the property of complementary angles in a right triangle, this shows that the ratios of \( \sin x° \) and \( \cos y° \) are connected through standard trigonometric identities:
\[ \sin x° = \cos y° \]